1. **State the problem:** Simplify and solve the expression $$(2x^2 - x)^2 - (2x - x^2)^2$$. 2. **Use the difference of squares formula:** Recall that $$a^2 - b^2 = (a - b)(a + b)$$. 3. **Identify $$a$$ and $$b$$:** Here, $$a = 2x^2 - x$$ and $$b = 2x - x^2$$. 4. **Apply the formula:** $$ (2x^2 - x)^2 - (2x - x^2)^2 = ((2x^2 - x) - (2x - x^2))((2x^2 - x) + (2x - x^2)) $$ 5. **Simplify each factor:** First factor: $$ (2x^2 - x) - (2x - x^2) = 2x^2 - x - 2x + x^2 = (2x^2 + x^2) - (x + 2x) = 3x^2 - 3x $$ Second factor: $$ (2x^2 - x) + (2x - x^2) = 2x^2 - x + 2x - x^2 = (2x^2 - x^2) + (-x + 2x) = x^2 + x $$ 6. **Rewrite the expression:** $$ (3x^2 - 3x)(x^2 + x) $$ 7. **Factor out common terms:** $$ 3x^2 - 3x = 3(x^2 - x) = 3x(x - 1) $$ $$ x^2 + x = x(x + 1) $$ 8. **Final factored form:** $$ 3x(x - 1) \times x(x + 1) = 3x^2 (x - 1)(x + 1) $$ 9. **Recognize difference of squares:** $$ (x - 1)(x + 1) = x^2 - 1 $$ 10. **Final simplified expression:** $$ 3x^2 (x^2 - 1) $$ This is the simplified form of the original expression. **Answer:** $$3x^2 (x^2 - 1)$$